festino

Description: "Festino", one of the syllogisms of Aristotelian logic. No ph is ps , and some ch is ps , therefore some ch is not ph . In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	festino.maj	$⊢ \forall x (φ \to \neg ψ)$
	festino.min	$⊢ \exists x (χ \land ψ)$
Assertion	festino	$⊢ \exists x (χ \land \neg φ)$

Proof

Step	Hyp	Ref	Expression
1		festino.maj	$⊢ \forall x (φ \to \neg ψ)$
2		festino.min	$⊢ \exists x (χ \land ψ)$
3		con2	$⊢ (φ \to \neg ψ) \to (ψ \to \neg φ)$
4	3	anim2d	$⊢ (φ \to \neg ψ) \to ((χ \land ψ) \to (χ \land \neg φ))$
5	4	alimi	$⊢ \forall x (φ \to \neg ψ) \to \forall x ((χ \land ψ) \to (χ \land \neg φ))$
6	1 5	ax-mp	$⊢ \forall x ((χ \land ψ) \to (χ \land \neg φ))$
7		exim	$⊢ \forall x ((χ \land ψ) \to (χ \land \neg φ)) \to (\exists x (χ \land ψ) \to \exists x (χ \land \neg φ))$
8	6 2 7	mp2	$⊢ \exists x (χ \land \neg φ)$

Metamath Proof Explorer

Theorem festino

Proof